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Prove that there is an additive function T: R > R (as defined in Exercise 37) that is
Chapter 2, Problem 39(choose chapter or problem)
Prove that there is an additive function T: R > R (as defined in Exercise 37) that is not linear. Hint: Let V be the set of real numbers regarded as a vector space over the field of rational numbers. By the corollary to Theorem 1.13 (p. 60), V has a basis ft. Let x and y be two distinct vectors in ft, and define /: ft > V by f(x) = y, f(y) = x, and f(z) = z otherwise. By Exercise 34, there exists a linear transformation Sec. 2.2 The Matrix Representation of a Linear Transformation 79 T: V -> V such that T(u) = f(u) for all u ft. Then T is additive, but for c = y/x, T(cx) ^ cT(x).
Questions & Answers
QUESTION:
Prove that there is an additive function T: R > R (as defined in Exercise 37) that is not linear. Hint: Let V be the set of real numbers regarded as a vector space over the field of rational numbers. By the corollary to Theorem 1.13 (p. 60), V has a basis ft. Let x and y be two distinct vectors in ft, and define /: ft > V by f(x) = y, f(y) = x, and f(z) = z otherwise. By Exercise 34, there exists a linear transformation Sec. 2.2 The Matrix Representation of a Linear Transformation 79 T: V -> V such that T(u) = f(u) for all u ft. Then T is additive, but for c = y/x, T(cx) ^ cT(x).
ANSWER:Step 1 of 3
Let 
